Determinant Properties: Row Operations & Multiplicative Rule

Effect of Row Swaps

Swapping two rows of a matrix multiplies its determinant by

-1

. If

B

is obtained from

A

by exchanging rows

i

and

k

, then

Determinant Row Swap

\det(B) = -\det(A)

Determinant row swap →

An immediate consequence is that any matrix with two identical rows has determinant zero. Swapping those two rows changes the sign of the determinant, yet the matrix itself is unchanged — the only number equal to its own negative is zero.

The same rule holds for columns: swapping two columns also flips the sign. This follows from transpose invariance, since swapping columns of

A

is the same as swapping rows of

A^T

, and

\det(A^T) = \det(A)

.

Pitfall: Ignoring the sign of a row swap

Swapping the rows of

\begin{pmatrix} 0 & 1 \\ 2 & 3 \end{pmatrix}

to reach the triangular

\begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}

and keeping its diagonal product gives

2

. The swap flips the sign: the determinant is

0 \cdot 3 - 1 \cdot 2 = -2

.

Determinant Notation

Notation

Determinant Notation

An operator with three spellings — a function name, a pair of bars, and a letter borrowed by three different jobs.

A

,

(a_{ij})

and

A^{T}

— matrix notation; the other bar species — absolute value and norms.

\det(A)

The determinant of A; det A

The operator spelling — parentheses optional, like

\sin x

:

\det A

and

\det(A)

are the same statement. Defined for square matrices only; the output is one scalar, sign included.

CasesParentheses turn mandatory on expressions:

\det(AB)

,

\det(A + B)

— and the notation is where the algebra shows its asymmetry:

\det(AB)

splits into a product in The Multiplicative Property below, while

\det(A+B)

splits into nothing at all.

Also written

\operatorname{det} A

upright in careful typesetting — an operator name, never italic letters

d \cdot e \cdot t

.

Do not confuseA matrix-valued operation.

\det

collapses the whole array to one number — writing matrix brackets around a determinant's result is a species error in reverse.

|A|

·

\begin{vmatrix} a & b \\ c & d \end{vmatrix}

The determinant of A, written with bars

The delimiter spelling: single bars around a matrix name, or straight bars replacing the matrix's own brackets — the one operator in elementary mathematics written as a pure delimiter. Between the bars sits the array; the bars themselves say “take its determinant”.

CasesThe bar family completes here:

|x|

on a scalar,

\|\mathbf{v}\|

on a vector,

|A|

on a matrix — and only this last one can be negative. The bars promise no non-negativity; a row swap above flips the sign inside them.

Also written

|A| = \det(A)

, freely interchangeable; the bars win inside computations,

\det

wins in prose and wherever

|{\cdot}|

is already busy with absolute values.

Do not confuseAbsolute value.

|\det(A)|

is a legitimate double-decker — the outer bars are the scalar absolute value of the possibly-negative number the inner operation produced; volume formulas use exactly this stack.

\Delta = \det(A)

Delta — the determinant, in the school tradition

Russian and European school texts reserve

\Delta

for the determinant of a system's coefficient matrix, with

\Delta_1, \Delta_2

(or

\Delta_x, \Delta_y

) for the numerator determinants of Cramer's rule:

x = \Delta_x / \Delta

.

CasesThe letter now holds three unrelated jobs across this site: the increment

\Delta x

, the discriminant

\Delta = b^2 - 4ac

, and this determinant. Field and context are the only disambiguators.

Also written

D

— the same tradition's other letter, colliding with the discriminant's

D

in the same texts; context carries the load twice over.

Do not confuse“Change in”. A student meeting

\Delta_x

in Cramer's rule after calculus reads “change in

x

” — here it is a fixed determinant with

x

's column replaced, no motion anywhere.

All linear algebra symbols →↑ Determinants

Effect of Row Scaling

Multiplying a single row of

A

by a nonzero scalar

k

multiplies the determinant by

k

. If

B

is obtained from

A

by replacing row

i

with

k

times row

i

, then

Determinant Row Scaling

\det(B) = k \cdot \det(A)

Determinant row scaling →

This is a single-row rule, not a whole-matrix rule. Scaling the entire matrix

A

by

k

means scaling every row, so

Determinant of Scalar Multiple

\det(kA) = k^n \det(A)

Determinant of scalar multiple →

for an

n \times n

matrix. A common error is to write

\det(kA) = k \det(A)

, forgetting that the scalar passes through each of the

n

rows independently.

Fact: Determinant

10^{-100}

, nowhere near singular

The matrix

0.1 I_{100}

has determinant

0.1^{100} = 10^{-100}

, one factor

0.1

from each of its

100

rows. Yet it only shrinks space evenly, and its inverse

10 I_{100}

is exact, so a tiny determinant alone does not mean a matrix is close to singular.

Factoring works in reverse as well: if every entry in some row shares a common factor, that factor can be pulled out in front of the determinant. For instance, if row

2

of a

3 \times 3

matrix is

(6, 12, 18)

, then

6

can be extracted to give a row of

(1, 2, 3)

and a factor of

6

multiplying the determinant. This often simplifies hand computations before beginning a cofactor or elimination approach.

A row of all zeros makes the determinant zero, since scaling that row by

0

gives

\det(A) = 0 \cdot \det(A') = 0

regardless of what

A'

looks like.

Effect of Row Addition

Adding a scalar multiple of one row to a different row leaves the determinant completely unchanged. If

B

is obtained from

A

by replacing row

i

with row

i

plus

c

times row

k

(where

i \neq k

), then

Determinant Row Addition

\det(B) = \det(A)

Determinant row addition →

This is the operation that does all the heavy lifting in Gaussian elimination, and it costs nothing in terms of the determinant. The reason traces back to the cofactor structure: the added row's contribution to the Laplace expansion along row

i

amounts to pairing entries from row

k

with cofactors from row

i

, which is a "wrong-row" expansion and always sums to zero.

Together, the three row-operation rules form a complete toolkit. Row addition is free, row scaling costs a known multiplicative factor, and row swapping costs a sign flip. Any sequence of these operations can be fully accounted for, which is what makes determinant computation via elimination both possible and efficient.

Computing Determinants via Row Reduction

The three row-operation rules convert Gaussian elimination into a determinant algorithm. The procedure is: reduce

A

to upper triangular form, record every row swap and every row scaling performed along the way, then compute the determinant of the triangular result as the product of its diagonal entries. Adjust by the accumulated sign flips and scale factors.

Worked Example

Computing Determinants via Row Reduction

A = \begin{pmatrix} 2 & 1 & -1 & 3 \\ 4 & 0 & 2 & 1 \\ -2 & 3 & 1 & 5 \\ 6 & 2 & 0 & -1 \end{pmatrix}

Subtract

2

times row

1

from row

2

, add row

1

to row

3

, and subtract

3

times row

1

from row

4

. All three are row-addition operations, so the determinant is unchanged:

Computing Determinants via Row Reduction

\begin{pmatrix} 2 & 1 & -1 & 3 \\ 0 & -2 & 4 & -5 \\ 0 & 4 & 0 & 8 \\ 0 & -1 & 3 & -10 \end{pmatrix}

Add

2

times row

2

to row

3

, and subtract

\frac{1}{2}

times row

2

from row

4

:

Computing Determinants via Row Reduction

\begin{pmatrix} 2 & 1 & -1 & 3 \\ 0 & -2 & 4 & -5 \\ 0 & 0 & 8 & -2 \\ 0 & 0 & 1 & -\frac{15}{2} \end{pmatrix}

Subtract

\frac{1}{8}

times row

3

from row

4

:

Computing Determinants via Row Reduction

\begin{pmatrix} 2 & 1 & -1 & 3 \\ 0 & -2 & 4 & -5 \\ 0 & 0 & 8 & -2 \\ 0 & 0 & 0 & -\frac{29}{4} \end{pmatrix}

No row swaps and no row scalings were used — only row additions. The determinant is the product of the diagonal:

Computing Determinants via Row Reduction

\det(A) = 2 \cdot (-2) \cdot 8 \cdot \left(-\frac{29}{4}\right) = (-32) \cdot \left(-\frac{29}{4}\right) = 232

Pitfall: Doubling a row to avoid fractions

Clearing the fraction with

R_4 \to 2R_4 - R_2

in place of

R_4 - \frac{1}{2}R_2

also doubles row

4

. The diagonal then ends in

-\frac{29}{2}

and the product is

464

. Dividing by that factor

2

gives

\det(A) = 232

.

Complexity

The reduction to triangular form requires roughly

\frac{2}{3}n^3

arithmetic operations. For a

10 \times 10

matrix this is about

670

operations; cofactor expansion on the same matrix would require on the order of

3.6

million. For anything beyond

4 \times 4

, row reduction is the only practical hand-computation method, and it is the standard numerical algorithm used by software.

The Multiplicative Property

For any two

n \times n

matrices

A

and

B

,

Determinant of Product

\det(AB) = \det(A) \cdot \det(B)

Determinant of product →

This is one of the most powerful structural facts about determinants. The proof splits into two cases. If

A

is singular, then

AB

is also singular (it cannot map onto all of

\mathbb{R}^n

if

A

already fails to), so both sides are zero. If

A

is invertible, it can be written as a product of elementary matrices, each corresponding to a single row operation. Since the determinant of each elementary matrix equals the factor by which that row operation multiplies the determinant, the result follows by chaining these factors together.

Corollaries

The multiplicative property generates several important identities at once. Since

AA^{-1} = I

and

\det(I) = 1

, the product rule gives

\det(A) \cdot \det(A^{-1}) = 1

, hence

Determinant of Inverse

\det(A^{-1}) = \frac{1}{\det(A)}

Determinant of inverse →

For any positive integer

k

:

Determinant of Power

\det(A^k) = (\det(A))^k

Determinant of power →

And since multiplication of determinants is commutative even when matrix multiplication is not:

The Multiplicative Property

\det(AB) = \det(A)\det(B) = \det(B)\det(A) = \det(BA)

Note that

AB

and

BA

generally differ as matrices, yet their determinants always agree.

A Non-Property

The determinant is not additive. In general,

\det(A + B) \neq \det(A) + \det(B)

. A quick counterexample: take

A = B = I_2

, so

\det(A) = \det(B) = 1

but

\det(A + B) = \det(2I_2) = 4

.

Identity	Formula	Note
Product	det(AB) = det(A) · det(B)	the central multiplicative rule
Inverse	det(A⁻¹) = 1 / det(A)	requires A invertible (det(A) ≠ 0)
Power	det(A^k) = (det A)^k	k a positive integer; extends to all integers if A invertible
AB vs BA	det(AB) = det(BA)	holds even when AB ≠ BA as matrices
Sum (NOT additive)	det(A + B) ≠ det(A) + det(B) in general	e.g., A = B = I₂ gives det(2I₂) = 4, not 1 + 1 = 2

The determinant of a product is the product of the determinants, a rule with no counterpart for sums.

Transpose Invariance

The determinant of a matrix equals the determinant of its transpose:

Determinant of Transpose

\det(A^T) = \det(A)

Determinant of transpose →

This single identity doubles the reach of every row-based property. Anything true about rows is automatically true about columns: swapping two columns flips the sign, scaling a column scales the determinant, and adding a multiple of one column to another leaves the determinant unchanged. Column expansion in the Laplace formula works precisely because the transpose identity converts it to a row expansion on

A^T

.

One way to see why the identity holds is through the permutation definition of the determinant. Transposing

A

replaces the permutation

\sigma

with its inverse

\sigma^{-1}

in each term of the sum. Since a permutation and its inverse have the same sign (both are even or both are odd), every term in the expansion is unchanged, and the total is the same.

Triangular and Diagonal Matrices

For an upper triangular, lower triangular, or diagonal matrix, the determinant is simply the product of the diagonal entries:

Determinant of Triangular

\det(A) = a_{11} \, a_{22} \cdots a_{nn}

Triangular matrix determinant →

This follows directly from cofactor expansion. For a lower triangular matrix, expanding along the first row leaves only the

(1,1)

entry (all others in the first row are zero), paired with the minor obtained by deleting row

1

and column

1

— which is again lower triangular. Repeating this reduction peels off one diagonal entry at a time until only the last entry remains.

As a special case, the identity matrix has determinant one:

Determinant of Identity

\det(I_n) = 1

Determinant of identity →

This property is what completes the row-reduction algorithm for computing determinants. Gaussian elimination produces an upper triangular matrix, and the determinant of that matrix is the product of its diagonal. Combined with the sign and scaling adjustments from the elimination steps, this gives

\det(A)

.

Only the diagonal product survives

Every term in the full expansion that strays off the diagonal picks up a zero somewhere and dies, leaving a single surviving product. This is why reducing a matrix to triangular form before taking its determinant is worth the effort: the elimination is cheap and the answer is then one multiplication chain. See the same shortcut applied on the determinant visualizer.

Row reduction and the determinant therefore work together rather than against each other, provided the swaps and scalings are tracked.

Block Triangular Matrices

A block triangular matrix is one that can be partitioned into square diagonal blocks with zero blocks either above or below:

Block Triangular Matrices

A = \begin{pmatrix} A_{11} & A_{12} \\ 0 & A_{22} \end{pmatrix} \quad \text{(block upper triangular)}

For such matrices, the determinant factors as

\det(A) = \det(A_{11}) \cdot \det(A_{22})

, and this extends to any number of diagonal blocks:

Block Triangular Determinant

\det(A) = \det(A_{11}) \cdot \det(A_{22}) \cdots \det(A_{kk})

Block triangular determinant →

The off-diagonal blocks

A_{12}

, etc., can contain anything — only the triangular placement of the zero blocks matters.

Example

Block Triangular Matrices

A = \begin{pmatrix} 3 & 1 & 0 & 0 & 0 \\ 2 & 5 & 0 & 0 & 0 \\ 0 & 0 & 1 & 4 & -2 \\ 0 & 0 & 0 & 3 & 1 \\ 0 & 0 & 0 & 0 & 7 \end{pmatrix}

This is block upper triangular with a

2 \times 2

block

A_{11} = \begin{pmatrix} 3 & 1 \\ 2 & 5 \end{pmatrix}

and a

3 \times 3

upper triangular block

A_{22} = \begin{pmatrix} 1 & 4 & -2 \\ 0 & 3 & 1 \\ 0 & 0 & 7 \end{pmatrix}

. The determinant is

\det(A_{11}) \cdot \det(A_{22}) = (15 - 2)(1 \cdot 3 \cdot 7) = 13 \cdot 21 = 273

.

Fact: Commuting blocks act like numbers

For square blocks of equal size,

\det\begin{pmatrix} P & Q \\ R & S \end{pmatrix} = \det(PS - QR)

whenever

RS = SR

, whatever

P

and

Q

are. Clearing

R

by block elimination leaves

\det(PS - QS^{-1}RS)

, and

RS = SR

cancels the

S^{-1}

.

This rule does not hold for general block matrices where the off-diagonal blocks are nonzero on both sides of the diagonal. The triangular structure is essential.

For a full block matrix

\begin{pmatrix} P & Q \\ R & S \end{pmatrix}

made of four square blocks of the same size, it is tempting to treat the blocks as numbers and apply the

2 \times 2

formula, giving

\det(PS - QR)

. Matrix products depend on the order of the factors, so there is no reason for this to work in general, and for most choices of blocks it gives the wrong value.

The Invertibility Equivalence

The determinant condenses the most fundamental structural question about a square matrix into a single test:

Determinant Invertibility Criterion

A \text{ invertible} \iff \det(A) \neq 0

This equivalence sits at the center of a web of conditions that are all mutually equivalent for an

n \times n

matrix

A

— the twelve below are either all true or all false.

Each of these conditions approaches invertibility from a different angle — rank, dimension, solvability, spectral theory — yet they all collapse to the same yes-or-no answer. The determinant is one entry in this list, but it is often the most efficient single computation for settling the question.

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Determinants · TFAE

One question, twelve answers

The determinant is one entry in this ring, not its centre. Every statement below implies every other, so any of them could have been the definition and the determinant chosen for convenience rather than priority.

12statements

$\det(A) \neq 0$ det
A is a product of elementary matricesdet
All eigenvalues of A are nonzerodet
$\operatorname{rank}(A) = n$ rank
The reduced row echelon form of A is $I_n$ rank
The columns of A are linearly independentspace
The rows of A are linearly independentspace
The columns of A span $\mathbb{R}^n$ space
The columns of A form a basis for $\mathbb{R}^n$ space
The null space of A is $\{\mathbf{0}\}$ space
$A\mathbf{x} = \mathbf{b}$ has exactly one solution for every $\mathbf{b}$ system
$A\mathbf{x} = \mathbf{0}$ has only the trivial solutionsystem

What the tagging exposesOnly three of the twelve mention the determinant at all. The other nine settle invertibility through rank, subspace structure, or solvability — none of which requires computing a determinant. That is worth holding onto, because for large

n

the determinant is among the most expensive routes to an answer the row reduction already gave you on the way.

det ×3rank ×2space ×5system ×2

One question, twelve answers·/linear-algebra/determinants/propertiesLearn Math Class

Summary of Determinant Properties

Across the sections above, each operation, transformation, or matrix structure has its own effect on the determinant. The table below collects every rule covered on the page into a single reference card — a one-stop lookup whenever a calculation requires recalling the right identity.

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Determinants · summary

What each operation does to the determinant

Eight properties, each linked to the section that establishes it. The three row operations at the top are the ones that make computation by reduction possible — everything below follows from them.

08properties

Unconditional2

iv

Row addition§ 3

R_i + cR_k \to R_i \;\Rightarrow\; \det(B) = \det(A)

The determinant is untouched.

Unconditional

Requires: $i \neq k$

The determinant is untouched. This is the operation Gaussian elimination leans on hardest — it can be applied freely without bookkeeping, which is what makes reduction a practical route to the determinant.

v

Transpose invariance§ 6

\det(A^{\mathsf{T}}) = \det(A)

Every statement about rows has a column counterpart, and neither is primary.

Unconditional

Requires: $A$ square

Every statement about rows has a column counterpart, and neither is primary. Cofactor expansion may be taken along any row or any column for the same reason.

Conditional5

i

Row swap§ 1

R_i \leftrightarrow R_k \;\Rightarrow\; \det(B) = -\det(A)

The only operation that changes the sign.

Sign flips

Holds when: any two distinct rows exchanged

The only operation that changes the sign. An even number of swaps restores it, which is why the sign of a permutation is well defined and why the determinant can be built from permutations at all. Swapping a row with itself changes nothing, consistent with the rule.

ii

Row scaling§ 2

kR_i \to R_i \;\Rightarrow\; \det(B) = k\det(A)

The determinant is linear in each row separately.

Conditional

Holds when: one row scaled by $k$
Common error: applying this to the whole matrix — see the next entry

The determinant is linear in each row separately. Scaling a single row by $k$ scales the determinant by $k$ ; scaling by $k = 0$ produces a zero row and a zero determinant, which is the same fact as a dependent row set.

iii

Scaling the whole matrix§ 2

\det(kA) = k^n\det(A)

Scaling every entry means scaling all n rows, so the factor applies n times.

Conditional

Holds when: $A$ is $n \times n$
Common error: writing $\det(kA) = k\det(A)$

Scaling every entry means scaling all $n$ rows, so the factor applies $n$ times. This is the single most common determinant error, and it comes from reading the row-scaling rule one line too broadly.

WitnessA = I₂, det(A) = 1
det(2A) = det([[2,0],[0,2]]) = 4 = 2² · 1, not 2

vi

Multiplicative property§ 5

\det(AB) = \det(A)\det(B)

The determinant is a homomorphism from matrix multiplication to real multiplication.

Conditional

Holds when: $A$ and $B$ both $n \times n$

The determinant is a homomorphism from matrix multiplication to real multiplication. Geometrically the volume scaling factors compose, which is the whole content of the identity. Note there is deliberately no corresponding rule for sums.

vii

Determinant of the inverse§ 9

\det(A^{-1}) = \dfrac{1}{\det(A)}

Falls straight out of the multiplicative property applied to AA^-1 = I.

Conditional

Holds when: $\det(A) \neq 0$

Falls straight out of the multiplicative property applied to $AA^{-1} = I$ . It also proves a singular matrix has no inverse — no real number multiplied by zero gives one.

Fails or undefined1

viii

Sum of matrices§ 5

\det(A + B) \neq \det(A) + \det(B)

Listed because its absence is otherwise inferred from silence.

Fails

Fails when: in general — no rule exists

Listed because its absence is otherwise inferred from silence. The determinant is linear in each row separately, which is a much weaker statement than being linear in the matrix, and the two are easy to conflate.

WitnessA = [[1,0],[0,0]], B = [[0,0],[0,1]]
det(A) + det(B) = 0, but det(A + B) = det(I₂) = 1

Unconditional

Conditional

Fails or undefined

Row swap§ 1Sign flips

R_i \leftrightarrow R_k \;\Rightarrow\; \det(B) = -\det(A)

Holds when: any two distinct rows exchanged

The only operation that changes the sign. An even number of swaps restores it, which is why the sign of a permutation is well defined and why the determinant can be built from permutations at all. Swapping a row with itself changes nothing, consistent with the rule.

Read the full section

Row scaling§ 2Conditional

kR_i \to R_i \;\Rightarrow\; \det(B) = k\det(A)

Holds when: one row scaled by $k$
Common error: applying this to the whole matrix — see the next entry

The determinant is linear in each row separately. Scaling a single row by

k

scales the determinant by

k

; scaling by

k = 0

produces a zero row and a zero determinant, which is the same fact as a dependent row set.

Read the full section

Scaling the whole matrix§ 2Conditional

\det(kA) = k^n\det(A)

Holds when: $A$ is $n \times n$
Common error: writing $\det(kA) = k\det(A)$

Scaling every entry means scaling all

n

rows, so the factor applies

n

times. This is the single most common determinant error, and it comes from reading the row-scaling rule one line too broadly.

WitnessA = I₂, det(A) = 1
det(2A) = det([[2,0],[0,2]]) = 4 = 2² · 1, not 2

Read the full section

Row addition§ 3Unconditional

R_i + cR_k \to R_i \;\Rightarrow\; \det(B) = \det(A)

Requires: $i \neq k$

The determinant is untouched. This is the operation Gaussian elimination leans on hardest — it can be applied freely without bookkeeping, which is what makes reduction a practical route to the determinant.

Read the full section

Transpose invariance§ 6Unconditional

\det(A^{\mathsf{T}}) = \det(A)

Requires: $A$ square

Every statement about rows has a column counterpart, and neither is primary. Cofactor expansion may be taken along any row or any column for the same reason.

Read the full section

Multiplicative property§ 5Conditional

\det(AB) = \det(A)\det(B)

Holds when: $A$ and $B$ both $n \times n$

The determinant is a homomorphism from matrix multiplication to real multiplication. Geometrically the volume scaling factors compose, which is the whole content of the identity. Note there is deliberately no corresponding rule for sums.

Read the full section

Determinant of the inverse§ 9Conditional

\det(A^{-1}) = \dfrac{1}{\det(A)}

Holds when: $\det(A) \neq 0$

Falls straight out of the multiplicative property applied to

AA^{-1} = I

. It also proves a singular matrix has no inverse — no real number multiplied by zero gives one.

Read the full section

Sum of matrices§ 5Fails

\det(A + B) \neq \det(A) + \det(B)

Fails when: in general — no rule exists

Listed because its absence is otherwise inferred from silence. The determinant is linear in each row separately, which is a much weaker statement than being linear in the matrix, and the two are easy to conflate.

WitnessA = [[1,0],[0,0]], B = [[0,0],[0,1]]
det(A) + det(B) = 0, but det(A + B) = det(I₂) = 1

Read the full section

The pattern worth carrying away: operations that preserve the row space leave the determinant alone or scale it predictably, and only a swap changes its sign. That is why row reduction can compute a determinant at all — every step’s effect is known in advance.

What each operation does to the determinant·/linear-algebra/determinants/propertiesLearn Math Class

Common Pitfalls

Three slips come up again and again when determinants are computed with row operations. Two are marked with a box where they happen in the sections above, and the third, pulling the scalar out of

kA

only once, is flagged in the row-scaling section. Here each is worked through in full, with the wrong and the right working side by side.

Ignoring the sign of a row swap

Rows are exchanged to bring a nonzero entry into a pivot position, and the reduction carries on as if the determinant had not changed. The diagonal product of the final triangular matrix is then reported as

\det(A)

, with the wrong sign whenever the number of swaps is odd.

Wrong working: for

A = \begin{pmatrix} 0 & 2 & 1 \\ 3 & 1 & 2 \\ 0 & 0 & 4 \end{pmatrix}

the first pivot is

0

, so rows

1

and

2

are swapped:

\begin{pmatrix} 3 & 1 & 2 \\ 0 & 2 & 1 \\ 0 & 0 & 4 \end{pmatrix}, \qquad 3 \cdot 2 \cdot 4 = 24

Right working: one swap was used, so the diagonal product is multiplied by

(-1)^1

, and

\det(A) = -24

.

Why it happens: a swap changes no entry, so it hardly feels like an operation on the matrix, and in a long reduction the count is easy to lose.

How to check: expand along a column with zeros. Column

1

of

A

has the single nonzero entry

3

, at position

(2,1)

, where the sign is

-

:

\det(A) = -3 \cdot \det\begin{pmatrix} 2 & 1 \\ 0 & 4 \end{pmatrix} = -3 \cdot 8 = -24

.

Pulling the scalar out once

A scalar multiplying the whole matrix is taken out of the determinant as a single factor,

\det(kA) = k \det(A)

. But

kA

scales every one of the

n

rows, and each row gives its own factor

k

.

Wrong working: for

A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}

,

\det(A) = 4 - 6 = -2

, and the single factor gives

\det(3A) = 3 \cdot (-2) = -6

.

Right working:

3A = \begin{pmatrix} 3 & 6 \\ 9 & 12 \end{pmatrix}

, so

\det(3A) = 36 - 54 = -18 = 3^2 \cdot (-2)

.

Why it happens: the row-scaling rule takes out exactly one factor of

k

, and multiplying the whole matrix by

k

is read as one scaling instead of

n

of them.

How to check: try

A = I_n

. The matrix

kI_n

is diagonal with

n

entries equal to

k

, so

\det(kI_n) = k^n

. For

n = 2

and

k = 2

this is

4

, the value in the non-property example, not

2

.

Doubling a row to avoid fractions

To keep the arithmetic in whole numbers, a row is replaced by a multiple of itself plus a multiple of another row, as in

R_4 \to 2R_4 - R_2

. That step is a row scaling and a row addition at once. The addition is free, but the scaling multiplies the determinant by

2

.

Wrong working: in the worked example, after the first round of eliminations, row

4

is

(0, -1, 3, -10)

and row

2

is

(0, -2, 4, -5)

. Replacing row

4

by

2R_4 - R_2

gives

(0, 0, 2, -15)

. Subtracting

\frac{1}{4}

of row

3

, which is

(0, 0, 8, -2)

, leaves

(0, 0, 0, -\frac{29}{2})

, and the diagonal product is

2 \cdot (-2) \cdot 8 \cdot \left(-\frac{29}{2}\right) = 464

Right working: the step

R_4 \to 2R_4 - R_2

doubled row

4

, so the product is

2 \det(A)

and

\det(A) = \frac{464}{2} = 232

. Using

R_4 \to R_4 - \frac{1}{2}R_2

, as the worked example does, gives

232

directly.

Why it happens: the step still looks like adding a multiple of one row to another, and the coefficient on the row being replaced is easy to overlook. Only that coefficient matters: in

R_i \to aR_i + cR_k

the determinant is multiplied by

a

, whatever

c

is.

How to check: for every step, look at the coefficient of the row being replaced. If it is not

1

, write it down, and divide the final diagonal product by the product of all such coefficients. The same bookkeeping covers dividing a pivot row by its pivot: halving row

1

of the worked example leaves a diagonal product of

116

, which has to be multiplied by

2

.

Determinant Properties FAQ

Why can $|A|$ be negative when $|x|$ and $\|\mathbf{v}\|$ cannot?

+

Because the bars mean something different here. On a scalar they take absolute value and on a vector they take length, both non-negative by construction, but on a matrix the same glyph denotes the determinant, which carries a sign. A single row swap flips it. That is also why

|\det(A)|

is a legitimate stack of two bar pairs.Read more →

Does $\det(A + B)$ equal $\det(A) + \det(B)$ ?

+

No, and nothing simpler replaces it: a determinant of a sum does not decompose at all. The asymmetry is worth noticing, since

\det(AB) = \det(A)\det(B)

splits perfectly for products. The operator is multiplicative but not additive, so borrowing the additive habit from ordinary algebra fails here.Read more →

In Cramer's rule, does $\Delta_x$ mean "change in $x$ "?

+

No. Despite the shared letter, nothing is changing:

\Delta_x

is a fixed determinant of the matrix with the

x

column replaced by the constants. The symbol holds three unrelated jobs across mathematics, as an increment in calculus, a discriminant in quadratics, and a determinant here. Only context separates them.Read more →

Properties of Determinants

How Row Operations and Algebra Shape the Determinant

Effect of Row Swaps

Determinant Notation

Determinant Notation

Effect of Row Scaling

Effect of Row Addition

Computing Determinants via Row Reduction

Worked Example

Complexity

The Multiplicative Property

Corollaries

A Non-Property

Transpose Invariance

Triangular and Diagonal Matrices

Block Triangular Matrices

Example

The Invertibility Equivalence

One question, twelve answers

Summary of Determinant Properties

What each operation does to the determinant

Common Pitfalls

Determinant Properties FAQ

Why can $|A|$ be negative when $|x|$ and $\|\mathbf{v}\|$ cannot?

Does $\det(A + B)$ equal $\det(A) + \det(B)$ ?

In Cramer's rule, does $\Delta_x$ mean "change in $x$ "?

Properties of Determinants

How Row Operations and Algebra Shape the Determinant

Effect of Row Swaps

Determinant Notation

Determinant Notation

Effect of Row Scaling

Effect of Row Addition

Computing Determinants via Row Reduction

Worked Example

Complexity

The Multiplicative Property

Corollaries

A Non-Property

Transpose Invariance

Triangular and Diagonal Matrices

Block Triangular Matrices

Example

The Invertibility Equivalence

One question, twelve answers

Summary of Determinant Properties

What each operation does to the determinant

Common Pitfalls

Determinant Properties FAQ

Why can ∣A∣|A|∣A∣ be negative when ∣x∣|x|∣x∣ and ∥v∥\|\mathbf{v}\|∥v∥ cannot?

Does det⁡(A+B)\det(A + B)det(A+B) equal det⁡(A)+det⁡(B)\det(A) + \det(B)det(A)+det(B)?

In Cramer's rule, does Δx\Delta_xΔx​ mean "change in xxx"?

Why can $|A|$ be negative when $|x|$ and $\|\mathbf{v}\|$ cannot?

Does $\det(A + B)$ equal $\det(A) + \det(B)$ ?

In Cramer's rule, does $\Delta_x$ mean "change in $x$ "?